Prediction of high-embankment settlement combining joint denoising technique and enhanced GWO-ν-SVR method

Qi Zhng ,Qin Su,b ,Zongyu Zhng ,Zhixing Deng ,De Chen,b,*

a School of Civil Engineering,Southwest Jiaotong University,Chengdu,China

b MOE Key Laboratory of High Speed Railway Engineering,Southwest Jiaotong University,Chengdu,China

Keywords: High embankment Settlement prediction Joint denoising technique Enhanced gray wolf optimizer Support vector regression

ABSTRACT Reliable long-term settlement prediction of a high embankment relates to mountain infrastructure safety.This study developed a novel hybrid model(NHM)that combines a joint denoising technique with an enhanced gray wolf optimizer (EGWO)-ν-support vector regression (ν-SVR) method.Highembankment field measurements were preprocessed using the joint denoising technique,which includes complete ensemble empirical mode decomposition,singular value decomposition,and wavelet packet transform.Furthermore,high-embankment settlements were predicted using the EGWO-ν-SVR method.In this method,the standard gray wolf optimizer (GWO) was improved to obtain the EGWO to better tune the ν-SVR model hyperparameters.The proposed NHM was then tested in two case studies.Finally,the influences of the data division ratio and kernel function on the EGWO-ν-SVR forecasting performance and prediction efficiency were investigated.The results indicate that the NHM suppresses noise and restores details in high-embankment field measurements.Simultaneously,the NHM outperforms other alternative prediction methods in prediction accuracy and robustness.This demonstrates that the proposed NHM is effective in predicting high-embankment settlements with noisy field measurements.Moreover,the appropriate data division ratio and kernel function for EGWO-ν-SVR are 7:3 and radial basis function,respectively.

1.Introduction

With the expansion of transportation demands,the construction of infrastructures in mountainous regions is on the rise(Mi??evi? and Vlastelica,2014;Zhu and Li,2020).High embankments,which serve as a solid foundation for upper structures and vehicles,have been widely used in mountainous regions due to the limitations of topographic conditions and line selection requirements(Vlastelica et al.,2016;Wu et al.,2019;Liu et al.,2020).The long-term settlement of a high embankment directly affects its service security(Vlastelica et al.,2018;Yao et al.,2018;Zheng et al.,2019).Therefore,a reliable long-term settlement prediction of high embankments is essential for high-embankment condition assessment and infrastructure safety.

Due to the external environmental and unforeseen factors,the field measurement of a high-embankment long-term settlement inevitably contains noise components (Chen et al.,2021).These components reduce the credibility of the measured data and compromise the prediction accuracy of the high-embankment long-term settlement.Hence,it is necessary to remove these noise components from the field measurements before prediction.Currently,studies on embankment field measurement denoising approaches can be categorized into three types: smoothing-based methods,filtering-based methods,and signal processing-based methods.The standard moving average method and Gaussianweighted moving average method are two widely used types of smoothing-based denoising approaches (Siddiqui et al.,2020a).However,the noise reduction effect of these methods highly depends on the adequacy of their interval settings(Tang et al.,2019).The most common filtering-based methods are the Savitzky-Golay filtering method (Schoefs et al.,2013),Butterworth filtering method (Lamas-Lopez et al.,2017),and Kalman filtering method(Tao et al.,2020).However,all of them have shortcomings.The appropriate filtering parameters of both the Savitzky-Golay filtering method and the Butterworth filtering method are determined by numerous trials,which are time-consuming and inefficient(Siddiqui et al.,2020a).The Kalman filtering method removes noises by recursively implementing “prediction-correction” calculations (Liao et al.,2022;Zhang et al.,2022a).However,the result credibility of the conventional Kalman filtering method is compromised when the prior knowledge of state and measurement noises is difficult to acquire (Zhang et al.,2022b).The Bayesianbased Kalman filter (Li et al.,2021) and its adaptive variant (Wu et al.,2022a) were proposed to archive this goal.In the meantime,they have been successfully used to reduce measurement noises in rolling bearing life prediction and geological surveys,respectively.Some signal processing-based methods such as the wavelet transform (WT) method (Chen et al.,2021) and empirical mode decomposition (EMD) method (Sungkono et al.,2014) have been applied to reduce the noise in measurement systems thanks to the advancement of signal processing techniques.However,the noise suppression ability of the WT method relates to the proper selections of the wavelet function,wavelet decomposition layers,etc.(Liu et al.,2018).Moreover,the WT method fails to finely denoise the high-frequency components of data,which results in the loss of effective information reserved in these high-frequency components (Zarei and Poshtan,2007).To overcome these limitations,the EMD method is adopted because of its adaptability in parameter settings,simple implementation,and preferable denoising effect (Bi et al.,2019).However,the major defect of the EMD method is modal aliasing,which can lead to decomposition errors and affect the denoising results (Han et al.,2019).The complete ensemble empirical mode decomposition (CEEMD)method (Yeh et al.,2010) was proposed as an improved variant of the EMD method,but opportunities remain to improve the denoising effectiveness.Therefore,the existing denoising methods,including the CEEMD method,remain insufficient to effectively reduce the noise in embankment field measurements,particularly those of high embankments.

Various studies have developed embankment settlement prediction methods,and they can be approximately classified into three types: analytical approaches,numerical calculations,and data-driven methods.Both analytical approaches (Yao et al.,2018)and numerical calculations (Zhang et al.,2020a) cannot readily obtain accurate settlement predictions,since determining the parameters of the two methods remains a major challenge in the prediction procedures (Siddiqui et al.,2022b).In contrast,datadriven methods,which include curve fitting methods (Li,2014),artificial neural network (ANN) models (Chen et al.,2021),and support vector regression(SVR)models(Javdanian et al.,2020),are employed to predict embankment settlement,due to their high computation accuracy and simple parameter settings.However,the prediction accuracy of the curve fitting method is significantly influenced by the initial time point determination (Li,2014).The accurate training of an ANN model usually requires a large amount of measured data,which is difficult to obtain in some field measurements (Luo et al.,2020).To overcome the limitations of the previous two prediction methods,the SVR model(Javdanian et al.,2020) is employed.With the outstanding ability to handle nonlinear settlement data,the SVR model provides a better prediction result using a small amount of training data.Additionally,the prediction accuracy of the SVR model is free of the initial time point determination.Therefore,the SVR model is more suitable for predicting embankment settlement.Nevertheless,the prediction performance of an SVR model is generally affected by the appropriate selection of its hyperparameters(Cheng et al.,2022).Hence,recent studies have concentrated on using various optimization algorithms to obtain the optimal hyperparameters of the SVR model to improve the model performance(Ali et al.,2021;Qiu and Wang,2017;Santos et al.,2021).

Previously,the appropriate hyperparameters of an SVR model were usually determined using a grid search method (Laref et al.,2019).However,the determination process of this approach is tedious and susceptible to trapping in a local optimum (Sun et al.,2021).To better determine the SVR model hyperparameters,metaheuristic optimization algorithms were introduced with their superior optimization performance(Mirjalili et al.,2014).However,all common metaheuristic optimization algorithms require tuning of their inherent parameters for better optimization results,such as the bat algorithm (BA),firefly algorithm (FA),and particle swarm optimization (PSO) (Laref et al.,2019).This requirement increases the computational complexity of determining hyperparameters,and may lead to prematurity when improper inherent parameters are used in these algorithms.

The gray wolf optimizer (GWO) algorithm (Mirjalili et al.,2014) has the advantage of few control parameters and global optimal solutions,and it can be considered a viable alternative to determine optimal hyperparameters.However,the standard GWO algorithm can easily fall into a local optimum due to the lack of wolf population diversity and an imbalance between exploitation and exploration during iteration (Heidari and Pahlavani,2017;Saxena et al.,2018).Various variants,most notably the mGWO algorithm (Mittal et al.,2016) and DE-GWO algorithm (Zhu et al.,2015),have been proposed to improve the optimization performance of the standard GWO algorithm.However,these improved GWO-based algorithms primarily improve either by maintaining the balance between exploitation and exploration or by maintaining the wolf population diversity.How to effectively escape trapping in a local optimum remains a major challenge (Nadimi-Shahraki et al.,2021).Thus,the SVR model must be combined with a better improved GWO optimization algorithm to accurately predict the settlement of an embankment,especially to predict the settlement of a high embankment that is prone to deformation.

To reliably predict a high-embankment settlement,this study established a novel hybrid model (NHM) based on the joint denoising technique and enhanced GWO-ν-SVR method,where ν is the parameter that controls the number of support vectors and training errors in SVR.Combining CEEMD with singular value decomposition (SVD) and wavelet packet transform (WPT),the joint denoising technique is promising for reducing the impact of noise on the accuracy of embankment settlement prediction.By optimizing the hyperparameters of a ν-SVR model with the enhanced gray wolf optimizer(EGWO)algorithm,the EGWO-ν-SVR method is proposed to predict high-embankment settlement.The main contributions of this study are as follows:(1)A joint denoising technique is provided to address the limitations of CEEMD processing in preprocessing high-embankment settlement data;(2)An EGWO algorithm is leveraged to mitigate the drawbacks of the standard GWO algorithm in ν-SVR model hyperparameter optimization;and (3) An NHM,which combines the joint denoising technique with the EGWO-ν-SVR method,is proposed to obtain accurate and robust prediction results.

The remainder of this study is organized as follows: Section 2 describes the joint denoising technique and EGWO-ν-SVR prediction method in detail.The results and analysis of two case studies are presented in Section 3.Section 4 investigates the effect of the data division ratio and kernel function on the EGWO-ν-SVR forecasting performance and prediction efficiency,and Section 5 provides the conclusion.

2.Methodology

2.1.Joint denoising technique

A joint denoising technique combines CEEMD processing,SVD processing,and WPT processing.The details of the joint denoising technique are described as follows.

2.1.1.Schemes of the joint denoising technique

Although CEEMD has significantly improved the noise filtering efficacy compared with EMD,it still has some drawbacks.To distinguish the noisy and effective IMFs decomposed by CEEMD,the correlation coefficients between each IMF and the original data are calculated(Sahu and Rai,2022).According to the conventional CEEMD processing procedure,the noisy IMFs must be removed,while the effective IMFs should be preserved to reconstruct the denoising data(Zheng et al.,2018;Yang et al.,2020).However,this removal can result in the loss of necessary details in the reconstructed data,since some useful information may be scattered in these noisy IMFs.Furthermore,the hybrid IMF,which is located after the boundary point between noisy IMFs and the effective IMFs,usually contains residual noise(Jia et al.,2015).To the best of our knowledge,conventional CEEMD processing does not include denoising measures for the hybrid IMF,which compromises the noise suppression effect in the reconstructed data.

Therefore,a joint denoising technique that combines CEEMD processing with SVD processing and WPT processing is proposed in this study,and the details of this technique are described as follows:

(1) Decompose the original data with CEEMD processing.Perform CEEMD processing on the original monitoring data using Eqs.(1)and(2)and obtain the IMFs and residue using Eq.(3).Then,calculate the correlation coefficient between each IMF and the original monitoring data.

(2) Determine the noisy IMFs,hybrid IMF,and effective IMFs.According to the literature (Jia et al.,2015),the IMF that corresponds to the first local minimum in the correlation coefficients is considered as the boundary point.The IMF at the point is the hybrid IMF,while the IMFs prior to the point are the noisy IMFs.Moreover,the remaining IMFs and residue can be set as the effective IMFs.

(3) Perform SVD processing and WPT processing.SVD processing is utilized to extract the features of the original monitoring data scattered in the noisy IMFs,and WPT is adopted processing to eliminate the residual noise in the hybrid IMF.The details of SVD processing can be found in the literature(Cheng et al.,2021;Zhang and Chen,2022),as can the details of WPT processing(Wu et al.,2022b;Zhang and Chen,2022).

(4) Reconstruct the denoising data.The denoising monitoring data is reconstructed by accumulating the noisy IMFs processed by SVD,hybrid IMF processed by WPT,and effective IMFs.

The flowchart of the joint denoising technique is shown in Fig.1.

Fig.1.Flowchart of the joint denoising technique.

2.1.2.Denoising performance verification

To illustrate the feasibility and effectiveness of the joint denoising technique,an embankment settlement simulation curve(Zhou et al.,2016) was selected for the tests,and it is shown as

whereS(t)is the real settlement simulation curve with noise,I(t)is the ideal settlement simulation curve without noise,N(t) is the added white noise that obeys a standard normal distribution,andt=300 is the duration time.

Then,the joint denoising technique was performed onS(t)using the procedures in Fig.1.The standard deviation of the positive and negative white noise added in CEEMD is 0.2,while the pair number of this noise is 100 (Colominas et al.,2014).After several trial calculations,the parameters for the WPT processing are taken as the db 6 wavelet function,rigrsure soft thresholding,and five-level decomposition.

The comparative results of the denoising settlement simulation curve,real settlement simulation curve,and ideal settlement simulation curve are shown in Fig.2.

Fig.2.Denoising comparison results: (a) Comparison between the denoising settlement simulation curve and real settlement simulation curve,and (b) Comparison between the denoising settlement simulation curve and ideal settlement simulation curve.

Fig.2 shows the ability of the joint denoising technique to reduce noise and extract information.In Fig.2a,compared with the noise-containing real settlement simulation curve,the denoising settlement simulation curve exhibits evident subsidence trend signatures of the embankment settlement.Furthermore,the denoising settlement simulation curve is much smoother than the real settlement simulation curve.The joint denoising technique can suppress noise when the original settlement curve is disturbed by noise.In Fig.2b,the denoising settlement simulation curve has a very high similarity with the ideal settlement simulation curve,and the correlation coefficient between the denoising settlement simulation curve and the ideal settlement simulation curve is 0.9993,demonstrating the ability of the denoising settlement curve to preserve details.In summary,the results indicate that the joint denoising technique can reliably eliminate the noise in the original embankment settlement data.

To validate the superiority of the joint denoising technique for embankment settlement data,several denoising methods were employed to perform contrastive analyses,including the CEEMD processing method and CEEMD-WPT method.The CEEMD processing method was executed by accumulating the effective IMFs,while the CEEMD-WPT method was performed by accumulating the effective IMFs from CEEMD processing and hybrid IMF after WPT processing.The signal-to-noise ratio (SNR) and mean square error(MSE)were adopted to evaluate the denoising performance of the above methods.Table 1 summarizes the performance evaluation results.

Table 1Performances of different denoising methods.

Table 1 reveals that the proposed joint denoising technique(CEEMD-SVD-WPT) has the highest SNR and lowest MES,which makes it an outstanding denoising method for embankment settlement data.Additionally,the CEEMD-WPT method performs slightly inferior to the joint denoising technique,which demonstrates the effectiveness of SVD processing on noisy IMFs.However,the CEEMD processing method has the lowest SNR and highest MES,which proves that the singular denoising method has defects in the embankment settlement data noise reduction.In conclusion,the proposed joint denoising technique has certain advantages in embankment settlement data noise reduction.

2.2.EGWO-ν-SVR prediction method

2.2.1.Theν-SVR model

According to the literature (Chang and Lin,2011),SVR models are classified into two types:ε-SVR and ν-SVR.The loss function is represented by ε in ε-SVR.Given the challenge of appropriately selecting the parameter ε in ε-SVR,ν-SVR is assigned to build the SVR model as follows:

wherewis the weight coefficient,θ（x）is the transformation function for the nonlinear function,andbis the model error.Then,Eq.(4)can be converted to the convex optimization problem as written in Eq.(5):

whereCis the error regularization coefficient,ν is the training error control parameter,ε is the insensitiveloss functioncoefficient,nis the volume ofthe supportvector,and ξiandarethe slack variables.

By introducing the Lagrange multipliers and kernel functionK（xi,xj）into Eq.(5),we can define Eq.(4) as

where αiandare nonzero Lagrange multipliers;and the radial basis function(RBF)(Cheng et al.,2017),which has a good ability in generation and learning,is employed as the kernel function.

Eqs.(4)-(6) demonstrate that accurate estimation using an ν-SVR model highly depends on the appropriate selection of its hyperparameters:the error regularization coefficientC,RBF kernel parameter γ,and training error control parameter ν.However,the previous method to determine these hyperparameters (Liu et al.,2021) is commonly time-consuming and low-precision,which undermines the reliability of the prediction results of an ν-SVR model.Therefore,GWO-based algorithms (Mirjalili et al.,2014),which are motivated by a new metaheuristic optimization,are adopted in this study to determine the optimal solutions for these hyperparameters.

2.2.2.EGWO algorithm

Inspired by the grouping foraging mechanism of gray wolves in nature,the standard GWO algorithm was proposed in 2014 as a novel metaheuristic optimization algorithm.When compared with other common metaheuristic optimization algorithms,the standard GWO algorithm is more efficient and reliable in obtaining global optimal solutions due to its unique leadership hierarchy.The major details of the standard GWO algorithm(Mirjalili et al.,2014)are described as follows:

(1) Encircling.When a preyXpis found,it is encircled by gray wolves using the following mathematical equations:

whereTmaxindicates the maximum iteration.

(2) Hunting.According to the leadership hierarchy in gray wolves,wolf population hunting is directed by the three leading wolves,i.e.α,β and δ wolves.The locations of the α,β and δ wolves at each iteration are updated by

Following that,the moving position of the wolf population at each iteration is determined by

However,the standard GWO algorithm has inherent shortcomings in obtaining the optimal hyperparameters (Zhang et al.,2020b;Guo et al.,2021).First,the initial population of each gray wolf is randomly generated,which results in a lack of population diversity at an early stage.Second,the linear decline convergence control factor (LDCCF) fails to satisfy the nonlinear adjustment requirement in the balance of exploitation and exploration.Finally,the new wolf population positions are updated by following the three leading wolves in each iteration,which reduces the wolf population diversity.The standard GWO algorithm may reach premature convergence and obtain suboptimal hyperparameters,which compromises the accuracy of the ν-SVR prediction model.

To improve the performance of the standard GWO algorithm,an enhanced version of the GWO named “EGWO” is proposed in this study.The EGWO includes cat chaotic mapping,a new nonlinear decline convergence control factor (NDCCF),and an individual learning-inspired population updating rule.The details of the EGWO are described as follows:

(1) Initial population generation using cat chaotic mapping.The chaotic mapping method helps strengthen the diversity of the initial wolf population (Jiang and Zhang,2021).Cat chaotic mapping outperforms the common chaotic mapping methods in maintaining the initial population diversity (Yu and Xu,2014).The dynamic equation of cat chaotic mapping is expressed as

wherexnis the chaotic vector at then-th iteration,ynis a random number at then-th iteration,and its range for the latter is (0,1).Mod1 is used to represent the fractional part of matrix multiplication results.Then,the initial population is generated by

whereis the position of thej-th wolf at thei-th iteration;AjandBjare the lower and upper limits of,respectively;andis the chaotic vector of.

(2) New nonlinear decline convergence control factor.The convergence control factor,which controls the balance between exploitation and exploration,contributes to obtaining the global optimum of the hyperparameters (Yang et al.,2019).To address the limitation of the LDCCF employed in the standard GWO,the NDCCF has been adopted(Wang et al.,2019).However,the value of NDCCF quickly decreases in the early iteration and slowly decreases in later iterations,which may cause the GWO to miss the global optimum in tuning the hyperparameters and increase the convergence time.Therefore,a new NDCCF is proposed in this study.The valuedecreasing speed of the proposed NDCCF is relatively slow in the early iteration to improve the exploration capacity and avoid missing the global optimum,while accelerating in the later iteration to increase the convergence speed.The proposed NDCCF is defined as

whereais the proposed NDCCF,tis the current number of iterations,andMax_iteris the maximum number of iterations.

(3) Improved wolf population updating rule based on an individual learning-inspired strategy.The wolf population updating rule in the GWO portrays the convergence approach to the optimal hyperparameters (Kaveh and Zakian,2018).Nevertheless,the updating strategy in the standard GWO may result in obtaining the local optimum by losing the diversity of the wolf population at each iteration.In response,an improved wolf population updating rule based on an individual learning-inspired strategy is proposed.Motivated by the concept of investigating the personal historical best position of each particle in PSO,the personal historical best position of each wolf is introduced into the existing wolf population,which updates the rule in every iteration to fortify its ability to capture the global optimum.The improved wolf population updating rule is written as

where（t）is the position of thei-th wolf at thet-th iteration;Xibestis the historical best position of thei-th wolf;Xi（t）（i=1,2,3）is the position of each leading wolf at thet-th iteration (Mirjalili et al.,2014);ωi（i=1,2,3）is the weight of each leading wolf and can be determined according to the literature(Chen et al.,2018);u1andu2are the updating coefficients,which can be set as 0.5 and 0.2 after several trials,respectively;andr1andr2are the random numbers in the range[0,1].

(4) Performance verification.To validate the effectiveness of the EGWO,the EGWO performance was analysed and compared with two widely used metaheuristic optimization algorithms,BA and FA,on 11 benchmarking functions from CEC 2014;Liang et al.(2013).Details of the eleven benchmarking functions are listed in Table 2.

Table 2Descriptions of the benchmarking functions.

The EGWO computation parameters are as follows: the wolf population is 30,the maximum iteration is 500,and the independent run time for each benchmarking function is 30 times.The EGWO-based computation process was performed in Windows 10 using MATLAB R2015b,a Core i5-7500H 3.40 GHz CPU,and 8 GB RAM.Two measures average(Avg)and standard deviation(Std)are calculated to evaluate the optimization performance of the three algorithms on each benchmarking function.The calculation results are listed in Table 3,where the results obtained by the BA and FA were obtained directly from the literature (Heidari and Pahlavani,2017) to ensure their validity.The best values are in bold font in Table 3.

Table 3Comparison results of the EGWO,BA and FA.

Table 3 reveals that the EGWO significantly outperforms the BA and FA on almost all benchmarking functions.For functionsf1(x)-f9(x)andf11(x),the EGWO has much better Avg values than BA and FA.Especially for functionsf5(x),f7(x),f10(x)andf11(x),the EGWObased Avg values reach the optima,while the Avg values of the BA and FA converge to the optima in functionsf10(x)andf11(x)only.The EGWO is more accurate in obtaining optima.Moreover,the EGWO has lower Std values in these functions than the BA and FA,which demonstrates that the EGWO has the greatest robustness in obtaining optima.Moreover,for functionf10(x),the EGWO performance is generally comparable to that of the BA and FA,although the EGWO has a slightly worse Std value than the FA.Therefore,the EGWO outperforms the other two algorithms in terms of optimization accuracy and robustness,and the improvements implemented in the EGWO are proven to be effective.

To investigate the optimization precision and robustness of the EGWO,the standard GWO and two of its state-of-the-art variants(mGWO (Mittal et al.,2016) and DE-GWO (Zhu et al.,2015)) were implemented to optimize the benchmarking functions in Table 2 for performance comparison.The DE-GWO optimization results were obtained from the literature (Zhu et al.,2015),while the mGWO optimization results were determined using the EGWO computation parameters and process.Table 4 lists the Avg and Std results of the four algorithms.The best values are in bold font in Table 4.

Table 4Comparison results of the EGWO,GWO,mGWO and DE-GWO.

Table 4 shows that the EGWO performs better than the standard GWO in almost all benchmarking functions.The EGWO has much better Avg and Std values than the standard GWO for functionsf1(x)-f9(x),while the EGWO has slightly worse Std values than the standard GWO on functionsf10(x) andf11(x).Moreover,the EGWO achieves the optima for twice as many functions as the standard GWO.Overall,the EGWO performance is superior to that of the standard GWO.

Table 4 also indicates that there is superior significance between the EGWO and the two variants of the standard GWO for most benchmarking functions,except for functionsf9(x)-f11(x).For functionf9(x),the EGWO has better Avg and Std values than mGWO but slightly worse Avg and Std values than DE-GWO,which indicates that the EGWO performs next only to DE-GWO onf9(x).Although the Std values of the EGWO are inferior to those of mGWO and DE-GWO on functionsf10(x) andf11(x),the EGWO reaches all optima.In conclusion,the EGWO obtains better optimization performance than the comparison algorithms.

2.2.3.Implementation of the EGWO-ν-SVR prediction method

Based on the EGWO algorithm and ν-SVR prediction model,the steps to propose a promising high-embankment settlement prediction method in this study are described as follows:

(1) Step 1: Data preprocessing.The denoising monitoring data,which has been processed by the joint denoising technique,is divided into a training set (70% of the denoising monitoring data) and a testing set (30% of the denoising monitoring data).To eliminate the influence of the monitoring data dimension on the prediction results,the training set and testing set are normalized using the method in the literature(Zhang and Hong,2021).

(2) Step 2:Parameter initialization.The parameters of the EGWO and hyperparameters of ν-SVR are initialized for optimization.For the EGWO,the initial parameters are as follows:the wolf population is 80 and the maximum number of iterations is 300.The ranges of the three hyperparameters of ν-SVR,i.e.C,γ and ν,are [0.01,100],[0.01,100],and [0.001,1],respectively.

(3) Step 3: Wolf population position initialization.Cat chaotic mapping,which is defined in Eqs.(15) and (16),is used to obtain the initial position of the wolf population.Then,the coordinate component of each wolf in the EGWO can be taken as the values of the ν-SVR hyperparameters.

(4) Step 4:Fitness value calculation and position updating.With the new nonlinear decline convergence control factor defined in Eq.(17),the fitness values(mean square error)of wolves in the training set are calculated to locate the positions of the three leading wolves at a certain iteration,and the position of the wolf population can be updated using the rule defined in Eq.(18).

(5) Step 5: Optimal hyperparameter determination.The iteration from steps 4 and 5 cannot be terminated until the maximum number of iterations is reached,and the coordinate components of the leading wolf α are simultaneously taken as the optimal hyperparameters of ν-SVR hyperparameters.

(6) Step 6: High-embankment settlement prediction method.A high-embankment settlement prediction method is proposed by substituting the optimal hyperparameters in Step 5 into ν-SVR.

(7) Step 7:Prediction performance evaluation.Some metrics are adopted to evaluate the performance of the proposed highembankment settlement prediction method on the testing set.

The flowchart of the proposed EGWO-ν-SVR prediction method is shown in Fig.3.

Fig.3.Flowchart of the enhanced GWO-ν-SVR prediction method.

2.3.Performance evaluation criteria

The correlation coefficient (R),root mean square error (RMSE),and mean absolute percentage error (MAPE) are widely used to evaluate the precision of a prediction method.The sum of squared prediction errors (SSPE) and squared prediction error (SPE) are generally employed to evaluate the robustness of a prediction method.These indices can be determined as follows:

wherexiandxpiare the actual value and predicted value of thei-th settlement data,respectively;andare the averages of the actual values and predicted values of all the settlement data,respectively;andnis the number of settlement data.

3.Case study

Two case studies,which contain different application scenarios of high embankments,are described in this section.In the first case study,a real-world settlement dataset of a heavy-haul railway high embankment in North China was adopted to compare the performance of the proposed NHM and commonly used prediction methods,the standard GWO-ν-SVR method,and the EGWO-ν-SVR method.In the second case study,an airport high-embankment settlement dataset from the literature (Yao et al.,2020) was used to compare the prediction performance of the proposed NHM and other methods,which included the aforementioned prediction methods and the method in the literature (Yao et al.,2020).The computation environment for different prediction methods was configured using MATLAB R2015b in Windows 10 with a Core i5-7500H 3.40 GHz CPU,and 8 GB RAM.The proposed NHM and aforementioned prediction methods had identical parameters in the two case studies.

3.1.Case study 1: high-embankment settlement prediction in a heavy-haul railway

3.1.1.Field data description and preprocessing

The real case addresses the Diandaigou-Nanping branch line of the Dazhun heavy-haul railway,which lies in Jungar Banner,Inner Mongolia,North China.The filling height of the branch line almost exceeds 25 m,which satisfies the requirement of a high embankment in a heavy-haul railway,and the slope ratio of the branch line is approximately 1:1.5.A geological survey shows that the strata of the embankment are as follows: silty soil,round grave soil,loess (),and mudstone (N2).The installation of hydrostatic levelling in the branch line began in August 2019 to measure both the left and right shoulder settlement of the embankment along the line.The measuring range of the hydrostatic leveling system is between 0 mm and 1000 mm,with a measurement accuracy of 0.001% of the measurement range.

Considering data availability and integrity,embankment section K13+527 was chosen for the study area,and its left shoulder settlement monitoring data were used for the dataset,where the data were sampled from August 10,2019 to November 18,2019.Fig.4 illustrates the layout of the chosen embankment section.

Fig.4.Layout of the study area.

Before performing settlement prediction,it is necessary to eliminate the noise that jams the settlement monitoring data(Huang and Wang,2018).The denoising settlement data can be reconstructed using the joint denoising technique.Fig.5 displays a comparison between the original data and denoising data.To quantitatively represent the noise level of the measurement data,the SNR of the original data was calculated.According to the literature (Jia et al.,2021),the SNR of the measurement data is 31.2135 dB.The joint denoising technique accurately separates the noise and restores the changing trend of the original data to the greatest extent,as shown in Fig.5.Furthermore,it helps to improve the precision of prediction results.Then,the original monitoring data were divided into a 70% training set and a 30% testing set,with the red dashed line showingt=70 d in Fig.5.

Fig.5.Comparison denoising effect using a heavy-haul railway high-embankment dataset.

3.1.2.Prediction results comparison and analysis

To analyse the predictive performance of the proposed NHM(the EGWO-ν-SVR method with joint denoising technique),we compare it with the EGWO-ν-SVR method and standard GWO-ν-SVR method.Other baseline methods are two commonly used approaches: the three-point method (a typical curve fitting method) and back-propagation neural network (BPNN) method (a typical ANN method).

The hyperparameters of the standard GWO-ν-SVR method were set to match those of the EGWO-ν-SVR method.The parameters in the three-point method were determined using the rule in the literature (Chen et al.,2016).The training period,training error,learning rate,and number of hidden layers are the hyperparameters of the BPNN method,which were set to 2×105,1 ×10-4and 0.1,respectively,and are determined by an empirical formula (Lee,2008).To ensure learning accuracy in the training set,we implemented the following iterative learning strategy: every four-settlement data were adopted for training to predict the following settlement data in turn when using the BPNN method,standard GWO-ν-SVR method,and EGWO-ν-SVR method.Table 5 lists the prediction performance metrics of these methods.

Table 5Forecasting performance metrics comparison based on a heavy-haul railway high-embankment settlement dataset.

Except for the proposed NHM,Table 5 illustrates that the EGWOν-SVR method performs best in prediction accuracy and robustness,with the highest value ofRand lowest values of RMSE,MAPE,SSPE and SPE.Thus,the EGWO-ν-SVR method has a better prediction effect than other alternative methods.Moreover,the EGWO-ν-SVR method outperforms the standard GWO-ν-SVR method,which demonstrates the advantages of the EGWO algorithm on the hyperparameter optimization and prediction performance of the ν-SVR model.

Furthermore,the proposed NHM,which combines the EGWO-ν-SVR method with the joint denoising technique,performs better than the single EGWO-ν-SVR method,which shows the positive effect of the joint denoising technique on improving the prediction performance.In general,the proposed NHM in this study has better prediction performance than other single prediction methods,although the dataset is heavily cluttered with noise.However,the BPNN method and three-point method are ranked at the bottom in terms of performance,indicating that these two methods are not suitable for precisely predicting heavy-haul high-embankment settlement.

To intuitively compare the performance of the five prediction methods,the prediction trend and residual results are plotted in Fig.6.In Fig.6a,the proposed NHM settlement prediction curve is closer to the field-measured settlement curve than those of the alternative prediction methods in both the training and testing sets.The results indicate that the proposed NHM achieves outstanding prediction accuracy.Moreover,the residual results in Fig.6b reveal that the residual value of the proposed NHM has the smallest fluctuation in the entire dataset,which demonstrates the prediction robustness of the proposed model.In summary,the proposed NHM is available for high-embankment settlement prediction in a heavy-haul railway.

Fig.6.Prediction comparison results based on a heavy-haul railway high-embankment dataset:(a)Trend comparison of the five methods,and(b)Residual comparison of the five methods.

3.2.Case study 2: high-embankment settlement prediction in an airport

3.2.1.Data resource and preprocessing

To further investigate the feasibility of the proposed NHM in high-embankment settlement prediction,a monitoring settlement data series of Lvliang Dawu airport (Yao et al.,2020) was adopted and ready for the prediction performance comparison.The Lvliang Dawu airport lies in the town of Dawu,Shanxi,North China.The maximum filling height of the airport is 85 m,and the majority of the filler is excavated loess.According to the geological survey,the original foundation strata beneath the filling embankment are as follows: Malan loess (),Lishi loess (),silty clay (),and sandy shale ().At the measurement point A,a digital level was installed on top of the airport fill embankment to monitor the postconstruction settlement of the airport.The measuring accuracy of the digital level is ±0.7 mm.Fig.7 depicts the location and monitoring layout of the airport.

Fig.7.Location and monitoring layout (Zhu and Li,2015) of the airport.

The monitoring project started from May 2010.The 153-d settlement monitoring data were first processed using the joint denoising technique,and the settlement trends before and after denoising are shown in Fig.8.The noise level of the measurement data can be evaluated with its SNR.The SNR value of the measurement data is 42.009 dB using the equation mentioned in the literature (Jia et al.,2021).

Fig.8.Comparison denoising effect using an airport high-embankment settlement dataset.

Fig.8 shows that the denoising settlement curve is highly consistent with the field monitoring settlement data curve,with some minor disturbances at the latter stage of the field monitoring settlement curve being effectively eliminated.This result also confirms the ability of noise reduction and detail preservation of the joint denoising technique in preprocessing the measured data with minor noise fluctuations.Subsequently,the red dashed line showingt=107 d in Fig.8 determines the training set and testing set of field monitoring settlements.

3.2.2.Prediction results comparison and analysis

The two commonly used methods as determined in Section

4.1.2,the literature-based method (Yao et al.,2020),the standard GWO-ν-SVR method,the EGWO-ν-SVR method,and the proposed NHM were implemented for comparative prediction using the field-measured settlement data.The prediction results of the literature-based method were obtained from the literature (Yao et al.,2020),and the iteration learning strategy for the training set in this section matched that in Section 4.1.2.The performance metric results of the six methods are listed in Table 6,and the predicted curves of the six methods are plotted in Fig.9.It should be noted that the best values of computational accuracy and robustness are in bold font in Table 6.

Table 6Forecasting performance metrics comparison based on an airport high embankment settlement dataset.

Fig.9.Prediction comparison results based on an airport high-embankment settlement dataset:(a) Trend comparison of the six methods,and (b) Residual comparison of the six methods.

Both Table 6 and Fig.9 demonstrate the superiority of the proposed NHM in prediction performance.According to Table 6,the proposed NHM achieves better prediction metric results than the other employed methods,including the literature-based method.There is only one exception that theR-value of the proposed NHM is slightly inferior to that of the literature-based method.These findings are based on the fact that the proposed NHM,which integrates the EGWO-ν-SVR method with the joint denoising technique,can more accurately and robustly estimate a nonlinear settlement time series in an airport high embankment than alternative methods.

Furthermore,Fig.9 illustrates that for the entire dataset,the prediction curve of the proposed EGWO-ν-SVR method generally results in the best predictions,and the error deviation from the measured settlement is smaller than that of the other four methods,particularly the literature-based method.Therefore,the appropriate prediction potential of the proposed NHM method on airport high-embankment settlement has been demonstrated.

4.Discussion

The denoising monitoring data is split into training and testing sets using the division ratio to build the EGWO-ν-SVR model and validate its performance(Tang and Na,2021).The size of a training set significantly influences model generation quality (Stojanovic et al.,2016).Therefore,how to appropriately determine the division ratio is related to the EGWO-ν-SVR model’s forecasting quality.Furthermore,the kernel function is critical in SVR nonlinear modelling,affecting its prediction ability(Xian and Che,2022).As a result,proper kernel function selection is beneficial for improving the EGWO-ν-SVR model’s prediction quality.This section determines the optimal division ratio and kernel function for the EGWO-ν-SVR model.For this,the influences of the division ratio and the kernel function on the forecasting performance and prediction efficiency of the EGWO-ν-SVR model are investigated.Moreover,the limitations and future works of this study are also discussed in this section.

4.1.Impact of division ratio on forecasting performance and prediction efficiency

The data division ratio balances the EGWO-ν-SVR model forecasting performance and prediction efficiency,and the prediction efficiency is measured by the computational time(Xue et al.,2021).A model with a large ratio for the training dataset outperforms a model with a small one in forecasting performance(Bardhan et al.,2022).Increasing the training dataset ratio,on the other hand,lengthens the computational time and decreases the prediction efficiency.Thus,comparative analyses were performed on the case study 1 settlement data to determine the appropriate data division ratio.The EGWO parameters and ν-SVR hyperparameters were matched with those in Section 3.2.3 during the analyses.The influence of the division ratio on the EGWO-ν-SVR forecasting performance was explored first.Four literature-based ratios(Stojanovic et al.,2016;Huang et al.,2022)and the ratio mentioned in this study were used to divide the settlement data into training and testing sets.Moreover,the last 20 samples from the abovementioned testing sets were assigned for forecasting performance comparison.The forecasting performance results based on these division ratios are listed in Table 7.The best values of computational accuracy and robustness are in bold font in Table 7.

Table 7Forecasting performance comparison based on different division ratios.

Table 7 illustrates that as the division ratio(the proportion of a training set size to a testing set size)increases,so do the EGWO-ν-SVR model computation accuracy and robustness.When the division ratio reaches 7:3,the model achieves the best forecasting performance,with the lowest values of RMSE,MAPE,SSPE,SPE,and the highest value ofR.When the division ratio exceeds 7:3,however,the computational accuracy and robustness of the model begin to generally decrease as the division ratio increases.It demonstrates that due to overfitting,the model forecasting performance has no positive correlation with an increasing division ratio.Therefore,a 7:3 division ratio is suitable for obtaining good forecasting performance.

Furthermore,the effect of the division ratio on the EGWO-ν-SVR prediction efficiency is discerned in Fig.10.

Fig.10.Prediction efficiency (measured by computational time) comparison based on different division ratios.

As shown in Fig.10,the computational time gradually increases as the division ratio increases from 1:1 to 7:3,while it sharply increases once the division ratio exceeds 7:3.It should be noted that the computation time with a 7:3 division ratio is approximately one-quarter that of a 4:1 division ratio.As previously stated,an appropriate division ratio is determined in terms of model prediction efficiency and forecasting performance.Thus,7:3 is the most appropriate division ratio for the EGWO-ν-SVR model,which demonstrates the rationality of the division ratio adopted in this study.

4.2.Impact of kernel function on forecasting performance and prediction efficiency

The establishment of a reliable SVR-based model requires finding an appropriate kernel function (Sun et al.,2021).An inappropriate kernel function could result in poor forecasting performance and low prediction efficiency (Ara et al.,2022).As for the proper kernel function in the EGWO-ν-SVR model,different kernel functions were chosen for comparing forecasting performance and prediction efficiency.These kernel functions are the linear kernel function,polynomial kernel function,sigmoid kernel function,and RBF kernel function that was adopted in this study(Liu et al.,2022).To conduct the comparison,the parameter settings of EGWO-ν-SVR were the same as those in Section 3.2.3.Moreover,the settlement data of Case Study 1 was divided into a training set (70% of the settlement data)and a testing set(30% of the settlement data),with the testing set used for comparison.The forecasting performance and prediction efficiency results based on these four kernel functions are depicted in Table 8 and Fig.11,respectively.The best values of computational accuracy and robustness are in bold font in Table 8.

Table 8Forecasting performance comparison based on different kernel functions.

Fig.11.Prediction efficiency (measured by computational time) comparison based on different kernel functions.

Both Table 8 and Fig.11 demonstrate the superiority of the RBF kernel function in forecasting performance and prediction efficiency.Table 8 shows that the EGWO-ν-SVR model with the RBF kernel function outperforms all others in computational accuracy and robustness.At the moment,all evaluation metrics perform optimally.This finding can be explained by the fact that the RBF kernel function has a strong learning ability in capturing the trend in settlement data (Li et al.,2021;Sun et al.,2021).That makes it available for predicting high-embankment settlements.As a result,the RBF kernel has an advantage in achieving accurate and credible forecasting.

Moreover,Fig.11 summarizes the change in prediction efficiency under different kernel functions used in the EGWO-ν-SVR model.In Fig.11,the computational time of the RBF kernel function is significantly lower than that of the other kernel functions.This result is based upon the fact that few parameters need to be determined in the RBF kernel function(Sun et al.,2021).It implies that the RBF kernel function helps to obtain optimal prediction efficiency.In conclusion,the RBF is the most appropriate kernel function for the EGWO-ν-SVR model,and it demonstrates the rationality of the kernel function adopted in this study.

4.3.Limitations and future works

Although the developed model provides improved accuracy and robustness of high-embankment settlement prediction,some limitations need to be further addressed.First,a trial-and-error method was utilized to determine WPT processing parameters,making the joint denoising technique time-consuming.Additionally,the RBF,which serves as the kernel function of the EGWO-ν-SVR model,still has room to improve the model prediction efficiency.Future research will concentrate on the adaptive parameter determination of WPT processing,and the construction of a hybrid SVR-based kernel function to accelerate model convergence speed.

5.Conclusions

In this study,an NHM to predict high-embankment settlement was developed by combining the joint denoising technique and the enhanced GWO-ν-SVR method.Using CEEMD processing,SVD processing,and WPT processing,the joint denoising technique effectively suppressed noise in the settlement monitoring data.The enhanced GWO-ν-SVR method was shown to successfully estimate the settlement behaviour of a high embankment.The feasibility and superiority of the developed NHM were fully illustrated by two case studies.Additionally,the effects of the data division ratio and kernel function on the EGWO-ν-SVR forecasting performance and prediction efficiency were studied.The main findings are summarized as follows:

(1) The joint denoising technique was found to be capable of eliminating the inherent noise fluctuation in settlement measurements,and greatly restoring the settlement measurement details.The joint denoising technique integrated the strengths of CEEMD processing,SVD processing,and WPT processing,and its noise suppression and detail preservation abilities were verified.Denoising settlement measurements can then markedly improve the precision of the prediction results.

(2) To address the shortcoming of the standard GWO algorithm in searching optimal hyperparameters,an enhanced version of the standard GWO algorithm,i.e.the EGWO algorithm,was proposed.The EGWO algorithm was developed based on cat chaotic mapping,a new nonlinear decline convergence control factor,and an individual learning-based population updating rule.The optimization performance of the EGWO algorithm was subsequently demonstrated using various benchmarking functions,and it was used to help tune the optimal hyperparameters of the ν-SVR model.

(3) The NHM,which couples the joint denoising technique with the EGWO-ν-SVR method,outperformed the EGWO-ν-SVR method,the standard GWO-ν-SVR method,and several other commonly used prediction methods in terms of both prediction accuracy and robustness.Furthermore,the NHM outperformed the literature-based method on the corresponding literature dataset.This result confirms the reliability of the proposed NHM in predicting high-embankment settlements in different application scenarios.

(4) The results of the comparative analyses indicated that the optimal data division ratio and kernel function for EGWO-ν-SVR were 7:3 and RBF,respectively.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We acknowledge the funding support from the National Natural Science Foundation of China (Grant No.51808462),the Natural Science Foundation Project of Sichuan Province,China (Grant No.2023NSFSC0346),and the Science and Technology Project of Inner Mongolia Transportation Department,China (Grant No.NJ-2022-14).We also sincerely thank Dr.Weizhen Fang from Analysis and Testing Center,Southwest Jiaotong University for his testing and analysis assistance.